Initially, the Federation was allied with the galactic empire of the Amstrons. Together they built the Gate of Creation, a wormhole to another universe, and sent a joint mission to explore it. The mission never returned and sent back a message claiming to have seen the face of the creator.
The disputes regarding whether this face resembled Amstrons or J'arrodic led to a long-term conflict (COMIC: The Infinite Astronaut) that came to be known as the Eternal Dogfight. It began in such distant past that even the scholars of the Time Lord Academy during the First Doctor's study there did not remember the origins of the conflict. After the J'arrodic homeworld was destroyed in the early stages of the war, the Federation made the J'arrodic mothership their base of operations. (COMIC: The Eternal Dogfight)